WEBVTT 00:00:01.710 --> 00:00:04.929 Hi, yesterday we imagined traveling a distance 00:00:04.929 --> 00:00:07.769 of one meter. First, we traveled halfway there, 00:00:07.889 --> 00:00:10.949 then half of what remained, and then half of 00:00:10.949 --> 00:00:13.929 what remained again. And then we ended with the 00:00:13.929 --> 00:00:16.750 question, if we continue that process forever, 00:00:17.089 --> 00:00:21.890 do we ever arrive at exactly one meter? Okay, 00:00:21.989 --> 00:00:24.829 in real life, the question eventually becomes 00:00:24.829 --> 00:00:27.969 a little silly. At some point, the distance remaining 00:00:27.969 --> 00:00:32.130 would be too small for us to measure. But mathematics 00:00:32.130 --> 00:00:35.630 lets us keep going. We can write the journey 00:00:35.630 --> 00:00:40.030 as an infinite sum. 1 half plus 1 fourth plus 00:00:40.030 --> 00:00:43.409 1 eighth plus 1 sixteenth plus, and then we do 00:00:43.409 --> 00:00:47.170 dot, dot, dot. Now, after any finite number of 00:00:47.170 --> 00:00:50.509 terms, we are still short of one. There's always 00:00:50.509 --> 00:00:54.969 something left. But the beauty is when we consider 00:00:54.969 --> 00:00:59.880 the entire infinite process. Mathematics tells 00:00:59.880 --> 00:01:03.799 us something remarkable. The sum is exactly one. 00:01:04.319 --> 00:01:09.379 Not approximately one, exactly one. Think about 00:01:09.379 --> 00:01:14.000 that. We've added infinitely many positive numbers 00:01:14.000 --> 00:01:17.480 together and ended up with something finite. 00:01:18.620 --> 00:01:23.239 We call this a convergent infinite series. The 00:01:23.239 --> 00:01:26.299 individual pieces keep getting smaller while 00:01:26.299 --> 00:01:29.480 the total gets closer and closer to a particular 00:01:29.480 --> 00:01:34.099 number. But what we said yesterday, when something 00:01:34.099 --> 00:01:38.400 matters, we fine -tune it. And mathematicians 00:01:38.400 --> 00:01:40.799 realize that saying an infinite series gets, 00:01:40.959 --> 00:01:44.920 quote, closer and closer is not quite good enough. 00:01:44.980 --> 00:01:49.980 What exactly does closer mean? How close is close 00:01:49.980 --> 00:01:54.400 enough? Not only that, even if two infinite processes 00:01:54.400 --> 00:01:58.239 have destinations, Do they approach them in the 00:01:58.239 --> 00:02:01.579 same way? Consider this beautiful series built 00:02:01.579 --> 00:02:04.719 from the odd numbers. We have 1 minus 1 third 00:02:04.719 --> 00:02:10.639 plus 1 fifth minus 1 seventh plus 1 ninth minus 00:02:10.639 --> 00:02:15.020 dot dot dot. We multiply the whole thing by 4 00:02:15.020 --> 00:02:20.219 and eventually it gives us pi. Now that's crazy. 00:02:20.419 --> 00:02:24.340 It's kind of amazing. It's beautiful. Anyway, 00:02:24.439 --> 00:02:27.900 nothing in the formula looks like a circle. It's 00:02:27.900 --> 00:02:30.840 just odd numbers, for goodness sakes, fractions, 00:02:30.960 --> 00:02:34.740 and alternating plus and minus signs. And somehow, 00:02:34.840 --> 00:02:39.419 someway, hidden inside it is pi. Now there is 00:02:39.419 --> 00:02:44.979 one problem. It is incredibly slow. If you calculate 00:02:44.979 --> 00:02:49.659 100 terms, you only get about two correct decimal 00:02:49.659 --> 00:02:53.750 places of pi accurate. And even after 10 ,000 00:02:53.750 --> 00:02:58.090 terms, you only have about 4. Yes, the series 00:02:58.090 --> 00:03:02.830 converges, but it takes its sweet time getting 00:03:02.830 --> 00:03:05.830 there. I'll compare that with another famous 00:03:05.830 --> 00:03:09.610 number, E. We can calculate E using this formula. 00:03:09.849 --> 00:03:14.710 1 plus 1 divided by 1 factorial plus 1 divided 00:03:14.710 --> 00:03:18.110 by 2 factorial plus 1 divided by 3 factorial 00:03:18.110 --> 00:03:22.080 plus 1 divided by 4 factorial. plus dot, dot, 00:03:22.080 --> 00:03:26.000 dot, where 3 factorial, for example, is 3 times 00:03:26.000 --> 00:03:30.960 2 times 1, 4 factorial is 4 times 3 times 2 times 00:03:30.960 --> 00:03:35.879 1, and so on. The bottom line is those denominators 00:03:35.879 --> 00:03:40.259 grow incredibly fast, and that means the terms 00:03:40.259 --> 00:03:45.280 become tiny very quickly, and the series closes 00:03:45.280 --> 00:03:52.000 in on E remarkably fast. So we have two infinite 00:03:52.000 --> 00:03:56.120 journeys, and both have destinations, but one 00:03:56.120 --> 00:04:00.159 crawls towards its destination while the other 00:04:00.159 --> 00:04:05.180 races there. So that is another layer of fine 00:04:05.180 --> 00:04:08.379 -tuning. In mathematics, it's often not enough 00:04:08.379 --> 00:04:13.419 to ask, does it work? We also ask, well, how 00:04:13.419 --> 00:04:16.939 well does it work? How quickly does it converge? 00:04:17.829 --> 00:04:21.329 How much computation does it require? How much 00:04:21.329 --> 00:04:24.769 accuracy do we gain with each additional step? 00:04:26.189 --> 00:04:29.670 So yesterday we asked where an infinite process 00:04:29.670 --> 00:04:33.129 is going. And today we fine -tune the question, 00:04:33.370 --> 00:04:36.389 how fast does it get there?